Session 004 — 2026-08-17 (resume after killed session; fork 2 complete)

Resume verification

Previous session was KILLED mid-work (python process died, taking the agent with it). STATE.md + git showed: last commit d5a952a (session 003, fork 1 E-010 done); uncommitted on disk — paper reorg, E-010 extended re-run, and E-011 (fork 2) with FAILING checks (V1/V2/V3 false, V5 true).

Diagnosis of the orphaned E-011 failure

• mtimes: data written 22:13, script edited 22:47 — the failing data came from an OLDER script revision; the fix was never re-run before the kill.
• Old run's λ₁ error had a 53-bit signature (~2e-16 abs). All precision paths in the NEW script probed clean to ~900 digits (mp.log/pi/euler vs flint at dps=1000; flint stieltjes worker strings; Pool pickle path).
• Found two real bugs left by the killed edit: 1. Hardcoded LAMBDA1_KNOWN was wrong from digit 22 — true λ₁ = 0.02309570896612103381431... (flint, 100 digits), not ...1055... 2. mp.dps getter typo → AttributeError (must be mp.mp.dps; cf. the known mp.dps setter bug in this env).
• Lesson (add to env lore): background Bash tasks are killed by the CLI timeout (~2 min default, 10 min max). Long runs must be launched DETACHED (subprocess.Popen, DETACHED_PROCESS|CREATE_NEW_PROCESS_GROUP) with a log file. flint.acb.stieltjes(k) scales ~k^3 (28s at k=2000, 3530 bits); full 0..2000 sweep ≈ 12.5 min on 14 workers.

Work

1. E-010 extended re-run verified (all 4 checks pass; a≤8, λ=−1, h_conv to 2.5e-3), re-registered in db, committed with paper reorg (b04e50a).
2. E-011 fixed and re-run: n_max=1000 validation PASS, then n_max=2000 detached run — ALL CHECKS PASS: - λ₁ err 4.31e-102; V2 worst dev ratio 0.125 (n≤116 vs E-008 zero side); - V3: λ_n > 0 for all n ≤ 2000; V5 rel dev 6.67e-705; - λ₂₀₀₀ = 5351.7595, Arias de Reyna refined ratio 1.00216. - Registered E-011 in db; result filed as R-001. - Runtime: stieltjes sweep 752s + V5 sweep 267s + sums ≈ 18 min.

Outcome

Fork 2 (task 3.2 continuation) COMPLETE: Li coefficients now computed to n=2000 from the arithmetic side (zero-free, BL Thm 2), cross-validated against the zero side (E-008). λ_n > 0 ∀ n ≤ 2000 [NUMERICAL-VALIDATED].

Fork 3 (task 3.3): de Bruijn–Newman Λ vs 2/3 density — CLOSED

• Literature check (D-001 Part III): 0 ≤ Λ ≤ 0.2 stands (Rodgers–Tao 2018/20 lower; Platt–Trudgian 2020 upper; no post-2020 improvement). Upper bounds on Λ are certified by RH-to-height + heat-flow dynamics.
• D-021: the density→Λ channel fails at the information-type level: (1) direction — the theorem lower-bounds the on-line fraction, Λ needs an upper bound on the complex fraction (no value of Λ contradicts the theorem); (2) extremal vs averaged — Λ is set by the first collision; (3) asymptotic vs explicit (Platt–Trudgian uses RH to 3·10¹²).
• E-012 (toy, all checks pass): two-body Rodgers–Tao collision times from E-002 gaps span 4.5 orders of magnitude (min 1.78e-4 vs median 0.108); full all-to-all backward ODE on 300 zeros: smallest-gap pair (j=298) collides first at t=−0.0229 vs two-body −0.0219 (4.8% correction).
• Filed: G-003 (route closed — do not retry with stronger density results). Open problem logged: a certificate that certifies zeros as COMPLEX (none known).

Fork 4 (thread T4): ξ′-analogue strength mechanism — CLOSED

• Question (D-013 item 4, G-001 salvage): why is the method stronger on ξ′ (0.85838/0.86864 vs 2/3)? Does the derivative break s₂/p?
• Answer (D-022): NO symmetry break — substitution is measurement-side, function-independent (G-001 salvage hypothesis corrected in place). The gain is data-side: ξ′ coefficients C(N;L_T) ≈ −Λ + (Λ·log)/L_T, Re(1/L_T) ~ 2/l, so the diagonal law has a PERFECT-SQUARE NODE D₁(s) ≈ s(1−2s)² — −Λ and (2/l)(Λ·log) cancel at prime scale p ~ √(T/2π). Pair integral (flat) 1/3 → 0.14162 = the entire gain.
• E-013 (all checks pass): R with kernel D₁ reproduces Remark 7.1 to 5e-6 (flat) / 2e-6 (quartic); node depth 0.089; kernel-s sanity 4/3.
• Portability verdict: NONE — the (Λ·log) partner is the derivative; log-weighting the ξ certificate = computing a ξ′ moment.
• Filed: R-002. Opened T4b (higher derivatives ξ⁽ᵏ⁾ — mechanism should iterate) + ξ′-world ceiling question (LawN256 analogue, cheap).
• ξ′-world nearly saturated (0.86864 vs Wu 0.86957 on-line-only).

ALL 4 FORKS COMPLETE (§8.4a) — gather thoughts with user next.

User chose option A (Q-S3 attack).

Q-S3 attack (Suzuki Cor 1.6) — E-014 + D-023

• Theory frame sharpened: W has exp type 2a + all-real zeros (Thm 1.5), e^{φ} ≠ 0 ⟹ zeros fixed; Cor 1.6 needs density COLLAPSE with a (outflow to ±∞ since zeros can't leave ℝ); θ only interlaces.
• Suzuki §7 read (heuristic: strong resolvent convergence D_{a,θ} → D under RH; spectrum Γ) — confirms the required spectral transition.
• E-014 (a ≤ 10, all checks pass): H1 zero count θ-independent (spread 1, argument principle) — de Branges interlacing verified; H2 v± boundary-localized ~0.80–0.83 (free e^{±x}-like) but NO thinning; H3 perfect π/a lattice to a=10, inflow continues, no γ-locking; H4 imag-axis read-off selects θ=π at EVERY a with perfect phase agreement (96/96) — residuals grow slowly with a (growth-dominated fit): candidate rule, not convergence evidence.
• D-023 verdict: Q-S3 sharply posed — transition must start at a ≫ 10; accessible window too small (sieve limit). SCALE limitation, not a wall. Queued Q-S3a (analytic: v± → A_∞^{−1}e^{±·}?), Q-S3b (backend redesign to a ~ 12–14), Q-S3c (Herglotz W′/W fine structure at θ=π).

Open options for the next stage

A1. Q-S3a/b/c (continuation, per D-023 §4) B. T4b scoping: higher derivatives ξ⁽ᵏ⁾ (node mechanism iterates?). C. ξ′-world bandwidth-one ceiling (LawN256 analogue — cheap). D. 3.2 continuation: push Li n past 2000 (detached runs) or asymptotic attack; optional Arb ball-arithmetic certification.